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Flat knot 6.1095

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,1,1,2,2,2,0,1,1,1,1,1,2,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1095']
Arrow polynomial of the knot is: 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.635', '6.639', '6.659', '6.727', '6.732', '6.735', '6.799', '6.1088', '6.1090', '6.1095', '6.1383']
Outer characteristic polynomial of the knot is: t^7+60t^5+55t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1095']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 2592*K1**4*K2 - 4496*K1**4 - 384*K1**3*K2**2*K3 + 576*K1**3*K2*K3 - 800*K1**3*K3 - 256*K1**2*K2**4 + 2912*K1**2*K2**3 - 8960*K1**2*K2**2 - 672*K1**2*K2*K4 + 10608*K1**2*K2 - 368*K1**2*K3**2 - 48*K1**2*K4**2 - 5296*K1**2 + 896*K1*K2**3*K3 - 416*K1*K2**2*K3 - 32*K1*K2**2*K5 + 6224*K1*K2*K3 + 632*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1392*K2**4 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 528*K2**2*K4 - 3144*K2**2 + 16*K2*K3*K5 - 1316*K3**2 - 224*K4**2 - 4*K5**2 + 4238
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1095']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.81562', 'vk6.81638', 'vk6.81642', 'vk6.81826', 'vk6.81834', 'vk6.82047', 'vk6.82226', 'vk6.82230', 'vk6.82336', 'vk6.82340', 'vk6.82532', 'vk6.82540', 'vk6.82997', 'vk6.83123', 'vk6.83125', 'vk6.83554', 'vk6.83556', 'vk6.83937', 'vk6.84086', 'vk6.84090', 'vk6.84529', 'vk6.84888', 'vk6.84896', 'vk6.85899', 'vk6.85901', 'vk6.86412', 'vk6.86416', 'vk6.86446', 'vk6.86448', 'vk6.88824', 'vk6.89756', 'vk6.89879']
The R3 orbit of minmal crossing diagrams contains:
The diagrammatic symmetry type of this knot is c.
The reverse -K is
The mirror image K* is
The reversed mirror image -K* is
The fillings (up to the first 10) associated to the algebraic genus:
Or click here to check the fillings

invariant value
Gauss code O1O2O3O4U1U5O6U2U3O5U4U6
R3 orbit {'O1O2O3O4U1U5O6U2U3O5U4U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U1O6U2U3O5U6U4
Gauss code of K* O1O2U3O4O5U2O6O3U1U4U5U6
Gauss code of -K* O1O2U3O4O5U1O3O6U2U4U5U6
Diagrammatic symmetry type c
Flat genus of the diagram 3
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 -1 1 2 -1 2],[ 3 0 1 2 3 1 3],[ 1 -1 0 1 1 0 2],[-1 -2 -1 0 0 -1 1],[-2 -3 -1 0 0 -2 0],[ 1 -1 0 1 2 0 2],[-2 -3 -2 -1 0 -2 0]]
Primitive based matrix [[ 0 2 2 1 -1 -1 -3],[-2 0 0 0 -1 -2 -3],[-2 0 0 -1 -2 -2 -3],[-1 0 1 0 -1 -1 -2],[ 1 1 2 1 0 0 -1],[ 1 2 2 1 0 0 -1],[ 3 3 3 2 1 1 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,-1,1,1,3,0,0,1,2,3,1,2,2,3,1,1,2,0,1,1]
Phi over symmetry [-3,-1,-1,1,2,2,1,1,2,2,2,0,1,1,1,1,1,2,0,1,0]
Phi of -K [-3,-1,-1,1,2,2,1,1,2,2,2,0,1,1,1,1,1,2,0,1,0]
Phi of K* [-2,-2,-1,1,1,3,0,0,1,1,2,1,1,2,2,1,1,2,0,1,1]
Phi of -K* [-3,-1,-1,1,2,2,1,1,2,3,3,0,1,1,2,1,2,2,0,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 5z^2+26z+33
Enhanced Jones-Krushkal polynomial 5w^3z^2+26w^2z+33w
Inner characteristic polynomial t^6+40t^4+31t^2
Outer characteristic polynomial t^7+60t^5+55t^3+4t
Flat arrow polynomial 4*K1**3 - 8*K1**2 - 2*K1*K2 - 2*K1 + 4*K2 + 5
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 2592*K1**4*K2 - 4496*K1**4 - 384*K1**3*K2**2*K3 + 576*K1**3*K2*K3 - 800*K1**3*K3 - 256*K1**2*K2**4 + 2912*K1**2*K2**3 - 8960*K1**2*K2**2 - 672*K1**2*K2*K4 + 10608*K1**2*K2 - 368*K1**2*K3**2 - 48*K1**2*K4**2 - 5296*K1**2 + 896*K1*K2**3*K3 - 416*K1*K2**2*K3 - 32*K1*K2**2*K5 + 6224*K1*K2*K3 + 632*K1*K3*K4 + 32*K1*K4*K5 - 32*K2**6 + 32*K2**4*K4 - 1392*K2**4 - 288*K2**2*K3**2 - 8*K2**2*K4**2 + 528*K2**2*K4 - 3144*K2**2 + 16*K2*K3*K5 - 1316*K3**2 - 224*K4**2 - 4*K5**2 + 4238
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {2, 5}, {1, 3}], [{5, 6}, {1, 4}, {2, 3}]]
If K is slice False
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